A unified and universal explanation for Lévy laws and 1/f noises

Iddo Eliazar*, Joseph Klafter

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Lévy laws and 1/f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals. The signals are considered to follow, statistically, a common - yet arbitrary - generic signal pattern which may be either stationary or dissipative. Each signal is considered to have its own random transmission amplitude and frequency. We characterize the amplitude-frequency randomizations which render the system output's stationary law and power-spectrum universal - i.e., independent of the underlying generic signal pattern. The classes of universal stationary laws and power spectra are shown to coincide, respectively, with the classes of Lévy laws and 1/f noises - thus providing a unified and universal explanation for the ubiquity of these classes of "anomalous statistics" in various fields of science and engineering.

Original languageEnglish
Pages (from-to)12251-12254
Number of pages4
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number30
StatePublished - 28 Jul 2009


  • Anomalous statistics
  • Poissonian randomizations
  • Shot noise
  • Universality


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